On the cut locus of closed sets

Let CC be a closed subset of a smooth manifold of dimension nn, MM, and let MM be endowed with a Riemannian metric of class C2C2. We study the cut locus of CC, cut(C). First, we show that cut(C) is a set of measure zero. Then, we assume that CC is the boundary of an open bounded set, Ω⊂MΩ⊂M (in particular, this assumption implies that cut(C)≠0̸.) We deduce that cut(C)∩Ω is invariant w.r.t. the (generalized) gradient flow associated with the distance function from the set CC. As a consequence of the invariance, we have that cut(C)∩Ω has the same homotopy type as the set ΩΩ. Furthermore, if MM is a compact manifold, then cut(C) has the same homotopy type as M∖CM∖C. Finally, we show that the closure of the cut locus stays away from CC if and only if CC is a manifold of class C1,1C1,1.